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Kaba
Posts:
289
Registered:
5/23/11


Leastsquares scaling
Posted:
Nov 11, 2012 1:55 PM


Hi,
Let
R in R^{d times n} P in R^{d times n}, and S in R^{d times d}, S symmetric positive semidefinite.
The problem is to find a matrix S such that the squared Frobenius norm
E = SP  R^2
is minimized. Geometrically, find a scaling which best relates the paired vector sets P and Q. The E can be rewritten as
E = tr((SP  R)^T (SP  R)) = tr(P^T S^2 P)  2tr(P^T SR) + tr(R^T R) = tr(S^2 PP^T)  2tr(SRP^T) + tr(RR^T).
Taking the first variation of E, with symmetric variations, and setting it to zero gives that
SPP^T + PP^T S = RP^T + PR^T
holds in the minimum point. One can rearrange this to
(SPP^T  RP^T)^T = (SPP^T  RP^T),
which says that SPP^T  RP^T is skewsymmetric. But I have no idea how to make use of this fact. Anyone?
 http://kaba.hilvi.org



